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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 12
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Articles

Sampling in shift-invariant spaces of functions with polynomial growth

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Pages 2536-2546 | Received 02 Mar 2012, Accepted 29 Oct 2012, Published online: 29 Nov 2012
 

Abstract

Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)} n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V ϕ ≔ {∑a n ϕ(t − n) : a n  ∈ ℂ}. In this space, there exist infinite functions with the same samples y n  = f(n). We show that polynomial growth conditions give the uniqueness: if y n has polynomial growth, there is a unique function of polynomial growth f ∈ V ϕ, satisfying f(n) = y n , n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y n  = f ∗ h(n).

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Acknowledgements

This work has been supported by the grant MTM2009–08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.

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